
Self-Test Problems and Exercises 385
7.7. Let X be the smallest value obtained when k num-
bers are randomly chosen from the set 1, ..., n.
Find E[X] by interpreting X as a negative hyper-
geometric random variable.
7.8. An arriving plane carries r families. A total of
n
j
of these families have checked in a total of j
pieces of luggage,
j
n
j
= r. Suppose that when
the plane lands, the N =
j
jn
j
pieces of luggage
come out of the plane in a random order. As soon
as a family collects all of its luggage, it immediately
departs the airport. If the Sanchez family checked
in j pieces of luggage, find the expected number of
families that depart after they do.
∗
7.9. Nineteen items on the rim of a circle of radius 1 are
to be chosen. Show that, for any choice of these
points, there will be an arc of (arc) length 1 that
contains at least 4 of them.
7.10. Let X be a Poisson random variable with mean λ.
Show that if λ is not too small, then
Var(
√
X) L .25
Hint: Use the result of Theoretical Exercise 4 to
approximate E[
√
X].
7.11. Suppose in Self-Test Problem 3 that the 20 peo-
ple are to be seated at seven tables, three of which
have 4 seats and four of which have 2 seats. If
the people are randomly seated, find the expected
value of the number of married couples that are
seated at the same table.
7.12. Individuals 1 through n, n > 1, are to be recruited
into a firm in the following manner: Individual 1
starts the firm and recruits individual 2. Individ-
uals 1 and 2 will then compete to recruit indi-
vidual 3. Once individual 3 is recruited, individu-
als 1, 2, and 3 will compete to recruit individual 4,
and so on. Suppose that when individuals 1, 2, ..., i
compete to recruit individual i + 1, each of them
is equally likely to be the successful recruiter.
(a) Find the expected number of the individuals
1, ..., n who did not recruit anyone else.
(b) Derive an expression for the variance of the
number of individuals who did not recruit
anyone else, and evaluate it for n = 5.
7.13. The nine players on a basketball team consist of 2
centers, 3 forwards, and 4 backcourt players. If the
players are paired up at random into three groups
of size 3 each, find (a) the expected value and (b)
the variance of the number of triplets consisting of
one of each type of player.
7.14. A deck of 52 cards is shuffled and a bridge hand
of 13 cards is dealt out. Let X and Y denote,
respectively, the number of aces and the number
of spades in the hand.
(a) Show that X and Y are uncorrelated.
(b) Are they independent?
7.15. Each coin in a bin has a value attached to it. Each
time that a coin with value p is flipped, it lands
on heads with probability p. When a coin is ran-
domly chosen from the bin, its value is uniformly
distributed on (0, 1). Suppose that after the coin is
chosen, but before it is flipped, you must predict
whether it will land on heads or on tails. You will
win 1 if you are correct and will lose 1 otherwise.
(a) What is your expected gain if you are not told
the value of the coin?
(b) Suppose now that you are allowed to inspect
the coin before it is flipped, with the result of
your inspection being that you learn the value
of the coin. As a function of p, the value of the
coin, what prediction should you make?
(c) Under the conditions of part (b), what is your
expected gain?
7.16. In Self-Test Problem 1, we showed how to use
the value of a uniform (0, 1) random variable
(commonly called a random number) to obtain the
value of a random variable whose mean is equal
to the expected number of distinct names on a list.
However, its use required that one choose a ran-
dom position and then determine the number of
times that the name in that position appears on
the list. Another approach, which can be more effi-
cient when there is a large amount of replication of
names, is as follows: As before, start by choosing
the random variable X
as in Problem 1. Now iden-
tify the name in position X, and then go through
the list, starting at the beginning, until that name
appears. Let I equal 0 if you encounter that name
before getting to position X,andletI equal 1 if
your first encounter with the name is at position X.
Show that E[mI] = d.
Hint: Compute E[I] by using conditional expecta-
tion.
7.17. A total of m items are to be sequentially dis-
tributed among n cells, with each item indepen-
dently being put in cell j with probability p
j
, j =
1, ..., n. Find the expected number of collisions
that occur, where a collision occurs whenever an
item is put into a nonempty cell.
7.18. Let X be the length of the initial run in a random
ordering of n ones and m zeroes. That is, if the first
k values are the same (either all ones or all zeroes),
then X Ú k.FindE[X].
7.19. There are n items in a box labeled H and m in a
box labeled T. A coin that comes up heads with
probability p and tails with probability 1 − p is
flipped. Each time it comes up heads, an item is
removed from the H box, and each time it comes
up tails, an item is removed from the T box. (If a
box is empty and its outcome occurs, then no items